A rotation system is a combinatorial way to encode an embedding of a graph on a surface. Instead of drawing an explicit embedding of a graph on the surface, the cyclic order of the edges around each vertex is instead recorded. The collection for all vertices of these cyclic orders (i.e., "rotations" at each vertex) then gives a rotation system. This cyclic order determines how edges are arranged locally and therefore determines the embedding up to homeomorphism.
A rotation system allows the faces of an embedding to be computed without any explicit drawing. However, rotation systems are not unique; different cyclic orders can produce different embeddings of the same graph.
To construct a rotation system, replace each edge by two directed half-edges and at each vertex specify a list of vertices based on which half-edge comes next when turning around the vertex in a counterclockwise direction.
A rotation system completely determines the vertices, edges, and faces of a graph, and therefore also the graph genus using Euler's formula.
